Zonotopal algebra and forward exchange matroids

Zonotopal algebra is the study of a family of pairs of dual vector spaces of multivariate polynomials that can be associated with a list of vectors X. It connects objects from combinatorics, geometry, and approximation theory. The origin of zonotopal algebra is the pair (D(X),P(X)), where D(X) denotes the Dahmen-Micchelli space that is spanned by the local pieces of the box spline and P(X) is a space spanned by products of linear forms. The first main result of this paper is the construction of a canonical basis for D(X). We show that it is dual to the canonical basis for P(X) that is already known. The second main result of this paper is the construction of a new family of zonotopal spaces that is far more general than the ones that were recently studied by Ardila-Postnikov, Holtz-Ron, Holtz-Ron-Xu, Li-Ron, and others. We call the underlying combinatorial structure of those spaces forward exchange matroid. A forward exchange matroid is an ordered matroid together with a subset of its set of bases that satisfies a weak version of the basis exchange axiom.Comment: 34 pages, 4 figures, minor corrections, same as journal version (up to layout

Development of an interactive computer graphics system with application to data fitting

The work reported in this thesis is organized into two parts. Part I presents a review study of the existing graphics facilities in terms of hardware and software (Chapter 2), interactive input techniques (Chapter 3) and the organization of graphics output processes and application data structures (Chapter 4). Finally, in Part I, a full account is presented concerning the development and implementation of the basic graphics software package LIGHT. Part II contains a detailed discussion of the implementation of several application programs which employ the basic graphics software developed in Part I. The applications cover the following problem areas: (1) Interpolatory Data Fitting (IDF); (2) Interactive Contour Tracing (ICT); (3) Triangular Mesh Generation (TMG). Finally, full program listings of the basic software and the application modules are given in the Appendices accompanying this thesis

Splines on polytopal complexes

This thesis concerns the algebra C^r(\PC) of $C^r$ piecewise polynomial functions (splines) over a subdivision by convex polytopes \PC of a domain $\Omega\subset\R^n$. Interest in this algebra arises in a wide variety of contexts, ranging from approximation theory and computer-aided geometric design to equivariant cohomology and GKM theory. A primary goal in approximation theory is to construct bases of the vector space C^r_d(\PC) of splines of degree at most $d$ on \PC, although even computing the dimension of this space proves to be challenging. From the perspective of GKM theory it is more important to have a good description of the generators of C^r(\PC) as an algebra; one would especially like to know the multiplication table for these generators (the case $r=0$ is of particular interest). For certain choices of \PC and $r$ there are beautiful answers to these questions, but in most cases the answers are still out of reach. In the late 1980s Billera formulated an approach to spline theory using the tools of commutative algebra, homological algebra, and algebraic geometry~\cite{Homology}, but focused primarily on the simplicial case. This thesis details a number of results that can be obtained using this algebraic perspective, particularly for splines over subdivisions by convex polytopes. The first three chapters of the thesis are devoted to introducing splines and providing some background material. In Chapter~\ref{ch:Introduction} we give a brief history of spline theory. In Chapter~\ref{ch:CommutativeAlgebra} we record results from commutative algebra which we will use, mostly without proof. In Chapter~\ref{ch:SplinePreliminaries} we set up the algebraic approach to spline theory, along with our choice of notation which differs slightly from the literature. In Chapter~\ref{ch:Continuous} we investigate the algebraic structure of continous splines over a central polytopal complex (equivalently a fan) in $\R^3$. We give an example of such a fan where the link of the central vertex is homeomorphic to a $2$-ball, and yet the $C^0$ splines on this fan are not free as an algebra over the underlying polynomial ring in three variables, providing a negative answer to a question of Schenck~\cite[Question~3.3]{Chow}. This is interesting for several reasons. First, this is very different behavior from the case of simplicial fans, where the ring of continuous splines is always free if the link of the central vertex is homeomorphic to a disk. Second, from the perspective of GKM theory and toric geometry, it means that the multiplication tables of generators will be much more complicated. In the remainder of the chapter we investigate criteria that may be used to detect freeness of continuous splines (or lack thereof). From the perspective of approximation theory, it is important to have a basis for the vector space C^r_d(\PC) of splines of degree at most $d$ which is `locally supported' in some reasonable sense. For simplicial complexes, such a basis consists of splines which are supported on the union of simplices surrounding a single vertex. Such bases are well known in the case of planar triangulations for $d\ge 3r+2$~\cite{HongDong,SuperSpline}. In Chapter~\ref{ch:LSSplines} we show that there is an analogue of locally-supported bases over polyhedral partitions, in the sense that, for $d\gg 0$, there is a basis for C^r_d(\PC) consisting of splines which are supported on certain `local' sub-partitions. A homological approach is particularly useful for describing what these sub-partitions must look like; we call them `lattice complexes' due to their connection with the intersection lattice of a certain hyperplane arrangement. These build on work of Rose \cite{r1,r2} on dual graphs. It is well-known that the dimension of the vector space C^r_d(\PC) agrees with a polynomial in $d$ for $d\gg 0$. In commutative algebra this polynomial is in fact the Hilbert polynomial of the graded algebra C^r(\wPC) of splines on the cone \wPC over \PC. In Chapter~\ref{ch:AssPrimes} we provide computations for Hilbert polynomials of the algebra $C^\alpha(\Sigma)$ of mixed splines over a fan $\Sigma\subset\R^3$, giving an extension of the computations in~\cite{FatPoints,TMcD,TSchenck09,Chow}. We also give a description of the fourth coefficient of the Hilbert polynomial of $HP(C^\alpha(\Sigma))$ where \Sigma=\wDelta is the cone over a simplicial complex $\Delta\subset\R^3$. We use this to re-derive a result of Alfeld-Schumaker-Whiteley on the generic dimension of $C^1$ tetrahedral splines for $d\gg 0$~\cite{ASWTet} and indicate via example how this description may be used to give the fourth coefficient in particular non-generic configurations. These computations are possible via a careful analysis of associated primes of the spline complex \cR/\cJ introduced by Schenck-Stillman in~\cite{LCoho} as a refinement of a complex first introduced by Billera~\cite{Homology}. Once the Hilbert polynomials which give the dimension of the spaces C^r_d(\PC) for $d\gg 0$ are known, one would like to know how large $d$ must be in order for this polynomial to give the correct dimension of the vector space C^r_d(\PC). Indeed the formulas are useless in practice without knowing when they give the correct answer. In the case of a planar triangulation, Hong and Ibrahim-Schumaker have shown that if $d\ge 3r+2$ then the Hilbert polynomial of C^r(\wPC) gives the correct dimension of C^r_d(\PC)~\cite{HongDong,SuperSpline}. In the language of commutative algebra and algebraic geometry, this question is equivalent to asking about the \textit{Castelnuovo-Mumford regularity} of the graded algebra C^r(\wPC). In Chapter~\ref{ch:Regularity}, we provide bounds on the regularity of the algebra $C^\alpha(\Sigma)$ of mixed splines over a polyhedral fan $\Sigma\subset\R^3$. Our bounds recover the $3r+2$ bound in the simplicial case. The proof of these bounds rests on the homological flexibility of regularity, similar in philosophy to the Gruson-Lazarsfeld-Peskine theorem bounding the regularity of curves in projective space (see~\cite[Chapter 5]{Syz})

Energy Calibration of the BaBar EMC Using the Pi0 Invariant Mass Method

The BaBar electromagnetic calorimeter energy calibration method was compared with the local and global peak iteration procedures, of Crystal Barrel and CLEO-II. An investigation was made of the possibility of {Upsilon}(4S) background reduction which could lead to increased statistics over a shorter time interval, for efficient calibration runs. The BaBar software package was used with unreconstructed data to study the energy response of the calorimeter, by utilizing the {pi}{sup 0} mass constraint on pairs of photon clusters

Structured Function Systems and Applications

Quite a few independent investigations have been devoted recently to the analysis and construction of structured function systems such as e.g. wavelet frames with compact support, Gabor frames, refinable functions in the context of subdivision and so on. However, difficult open questions about the existence, properties and general efficient construction methods of such structured function systems have been left without satisfactory answers. The goal of the workshop was to bring together experts in approximation theory, real algebraic geometry, complex analysis, frame theory and optimization to address key open questions on the subject in a highly interdisciplinary, unique of its kind, exchange

Image analysis using multiscale boundary extraction algorithm

The complete analysis and interpretation of the information in image data is a complex process. This dissertation presents 3 major contributions to image analysis, namely, global multiscale detection, local scale analysis, and boundary extraction. Global scale analysis is related to identification of the various scales presented in the image. A new approach for global scale analysis is developed based on the differential power spectrum normalized variance ratio (DPSNVR). The DPSNVR is the ratio of the second order normalized central moment of the power spectrum of the image to that of the multiscale differential mask. Local maxima in DPSNVR graph directly indicate the global scales in the image. Local scale analysis performs a more detailed analysis of the edges to eliminate effects of blurring. A method based on mutilscale feature matching has been proposed. Details obtained at all scales are treated using a scale invariant normalization scheme. Besides local scale analysis, a multiscale data fusion algorithm has been implemented which leads to the new concept of multiple scale differential masks. The multiple scale differential mask generated using a range of scale values possesses the remarkable shape preservation property which makes it superior to traditional multiscale masks. Finally the complete sequential boundary extraction algorithm based on particle motion in a velocity field is presented. The boundary extraction algorithm incorporates edge localization, boundary representation, and automated selection of boundary extraction parameters. The global scale analysis techniques in conjunction with the boundary extraction algorithm provide a multiscale image segmentation algorithm

New strategies for curve and arbitrary-topology surface constructions for design

This dissertation presents some novel constructions for curves and surfaces with arbitrary topology in the context of geometric modeling. In particular, it deals mainly with three intimately connected topics that are of interest in both theoretical and applied research: subdivision surfaces, non-uniform local interpolation (in both univariate and bivariate cases), and spaces of generalized splines. Specifically, we describe a strategy for the integration of subdivision surfaces in computer-aided design systems and provide examples to show the effectiveness of its implementation. Moreover, we present a construction of locally supported, non-uniform, piecewise polynomial univariate interpolants of minimum degree with respect to other prescribed design parameters (such as support width, order of continuity and order of approximation). Still in the setting of non-uniform local interpolation, but in the case of surfaces, we devise a novel parameterization strategy that, together with a suitable patching technique, allows us to define composite surfaces that interpolate given arbitrary-topology meshes or curve networks and satisfy both requirements of regularity and aesthetic shape quality usually needed in the CAD modeling framework. Finally, in the context of generalized splines, we propose an approach for the construction of the optimal normalized totally positive (B-spline) basis, acknowledged as the best basis of representation for design purposes, as well as a numerical procedure for checking the existence of such a basis in a given generalized spline space. All the constructions presented here have been devised keeping in mind also the importance of application and implementation, and of the related requirements that numerical procedures must satisfy, in particular in the CAD context

Algebraic Approaches for Constructing Multi-D Wavelets

Wavelets have been a powerful tool in data representation and had a growing impact on various signal processing applications. As multi-dimensional (multi-D) wavelets are needed in multi-D data representation, the construction methods of multi-D wavelets are of great interest. Tensor product has been the most prevailing method in multi-D wavelet construction, however, there are many limitations of tensor product that make it insufficient in some cases. In this dissertation, we provide three non-tensor-based methods to construct multi-D wavelets. The first method is an alternative to tensor product, called coset sum, to construct multi-D wavelets from a pair of $1$-D biorthogonal refinement masks. Coset sum shares many important features of tensor product. It is associated with fast algorithms, which in certain cases, are faster than the tensor product fast algorithms. Moreover, it shows great potentials in image processing applications. The second method is a generalization of coset sum to non-dyadic dilation cases. In particular, we deal with the situations when the dilation matrix is \dil=p{\tt I}_\dm, where $p$ is a prime number and {\tt I}_\dm is the \dm-D identity matrix, thus we call it the prime coset sum method. Prime coset sum inherits many advantages from coset sum including that it is also associated with fast algorithms. The third method is a relatively more general recipe to construct multi-D wavelets. Different from the first two methods, we attempt to solve the wavelet construction problem as a matrix equation problem. By employing the Quillen-Suslin Theorem in Algebraic Geometry, we are able to build \dm-D wavelets from a single \dm-D refinement mask. This method is more general in the sense that it works for any dilation matrix and does not assume additional constraints on the refinement masks. This dissertation also includes one appendix on the topic of constructing directional wavelet filter banks